Overview

• We want to forecast a time series.
  • returns, interest rates, oil price, cash flows, …
• Possible models:
  • ARIMA (autoregressive integrated moving average)
  • exponential smoothing
  • neural networks
• We’ll do AR models.

Autocorrelations

• Autocorrelation = correlation of a variable with its lagged value
• First-order autocorrelation \(=\text{corr}(x_t, x_{t+1})\)
• \(p\)–th order autocorrelation \(=\text{corr}(x_t, x_{t+p})\)
  • E.g., \(p=12\) in monthly data or \(p=4\) in quarterly data because of seasonality

Autocorrelation of HML

• Ask Julius to use pandas-datareader to download the monthly Fama-French factors from Ken French’s data library.
• Ask Julius to produce a plot of the autocorrelation function (acf) for HML.

Autoregressions

• An autoregression is a regression of a variable on its own lags: \[x_t = \alpha + \beta_1x_{t-1} + \cdots + \beta_px_{t-p} + \varepsilon_t\]
• Ask Julius to fit an AR(1) for HML.
• Ask Julius to use the AR(1) model to forecast HML for the next 12 months and to plot the last 2 years and the forecasts.

More examples

Ask Julius to plot the ACF, fit an AR model, and forecast the following:

• Percent change in crude price: Ask Julius to use pandas-datareader to download crude oil prices starting in 1980 from FRED, to downsample to end-of-month, and to compute percent changes.

• Change in Treasury yields: Ask Julius to use pandas-datareader to download 10-year Treasury yields starting in 1980 from FRED, to downsample to monthly, and to compute changes.

Vector Autoregression

• Forecast related variables based on their mutual lags
• Example: does this month’s SMB return forecast next month’s HML return?
• Ask Julius to run a VAR(1) on the Fama-French factors and provide a summary of the results.